The Mathematics Series

Focuses on four essential aspects of effective teaching: (1) deepening understanding of mathematics content for teaching; (2) understanding student thinking; (3) effective formative assessment strategies; and (4) developing an environment that fosters problem solving.

• Foundations of Effective Mathematics Teaching
  Highlights the rationale and pedagogy underlying effective practice. Continuous improvement of both teaching and learning is stressed.
• Effective Questioning in the Mathematics Classroom
  Presents ways that questioning makes student thinking transparent and helps deepen their understanding and extend their comprehension. Differences between one-to-one and group discussions are stressed.
• Formative Assessment in the Mathematics Classroom
  Offers rationale and strategies for formative assessments that elicit student understanding and inform teaching practice. Integrating formative assessments with standards and high-stakes testing is stressed.

Uses work with a “broken” calculator to identify creative problem-solving strategies and use these to support students as they build understanding of the meaning of mathematics, gain number sense, and compute with fluency.

• Problem Solving in Mathematics
  Offers views of various problem-solving strategies and the teacher’s role in creating an environment that supports problem solving. Using questioning and discussions to understand student thinking is stressed.

• Broken Calculator
  Uses exploration of a broken calculator to promote number sense, computational fluency, and problem solving. Student communication about the strategies and solutions is stressed.

Explores the Big Ideas of two core concepts in elementary math—division with remainders and the magnitude of fractions—and the research-based strategies to promote student understanding of each.

• Division with Remainders
  Provides insight into the multiple meanings of division and how to help students contextualize and understand division problems. Effective questioning strategies are stressed.

• The Magnitude of Fractions
  Explores effective ways to help students represent and compare fractions as numbers with magnitude, not just as parts of a whole. Proportional (multiplicative) reasoning is stressed.

Provides insights into understanding and teaching the foundational concepts of algebra—patterns, equivalence, order of operations, variables, and use of algebraic symbols and representations.

• Pan Balance Equations
  Deepen understanding of the concepts of equivalence, order of operations, and appropriate use of algebraic symbols. Effective questioning strategies to promote and support student learning are stressed. 

• Patterns and Functions
  Explore using patterns to identify functions, predict items in a sequence, and represent these as pictures, charts, and linear equations. Correct use of mathematics vocabulary is stressed.

Promotes geometric thought through application of the Van Hiele framework, strategic questioning, inductive and deductive reasoning, and building understanding of the ways in which both concepts and procedures deepen thinking.

• 2D and 3D Figures
  Uses strategies that include reasoning, the vocabulary of mathematics, and a four-stage version of the Van Hiele framework to develop student understanding of geometry. Students’ use of reasoning and proof are stressed.

• Calculating the Area of a Triangle
  Promotes the use of both concepts and procedures as problem-solving approaches to help students understand effective ways to find areas of nonstandard figures. The use of journaling and appropriate mathematics vocabulary to support understanding is stressed.

Focuses on helping students use measures of center to describe data sets and to order and organize the data to make predictions that sometimes defy the notion of “fairness.”

• Measures of Center
  Explores how to help students organize, represent, and describe data and data landmarks and use these to inform initial thinking about a situation. Scaffolding instruction for English learners and use of questioning are stressed in this module.

• Using Data to Make Predictions
  Deepens understanding of probability by exploring concepts of social vs. mathematical fairness, causality, and the relationship between sample size and predictions. Use of activities to contextualize the concepts and promote understanding is stressed.

Explores how a teacher uses an engaging activity to help students understand proportional reasoning and apply it to concepts in geometry, measurement, operations, and algebra.

• Proportional Reasoning in the Middle Grades
  Promotes application of proportional reasoning to problem solving and to connecting content across the mathematics curriculum. The four levels of proportional reasoning and their application to specific types of problems are stressed.

Promotes the use of measures of center to interpret relationships within and among data sets by exploring and interpreting patterns.

• Data Analysis
  Focuses on understanding the ways students think about measures of center, the relationships among data sets, and ways to represent the data to support deep understanding. Use of inductive reasoning to infer relationships is stressed. 

Explores the relationship between proportional reasoning and algebraic thinking and offers strategies to develop proportional reasoning through understanding student thinking about real-world problems.

• Proportional Reasoning
  Helps teachers deepen their understanding of and address the predictable challenges of helping students acquire and apply proportional reasoning to understand the multiple meanings of proportions and their applications to algebra. Understanding and advancing student thinking about proportional reasoning is stressed.

Grounds the study of linear functions and transformations of linear functions within the context of real-world applications to promote the concepts of rate of change, multiple representations, and the graphic and symbolic form of functions as objects for operations.

• Linear Functions
  Focuses on functions as the core objects of algebra by representing, evaluating, and manipulating them in multiple ways and in various formats that represent real-world contexts. Observing and understanding student thinking is stressed.

• Transformations of Linear Functions
  Focuses on the relationship between graphic and symbolic forms of functions as core concepts of algebra—as objects that can be operated upon. The emphasis in this module is on helping students shift from thinking of functions as processes to thinking of functions as objects. 

Focuses on the relationship between equations and functions by promoting a view of equations as functions to which values have been assigned. It also focuses on the context of real-world applications as a way of understanding linear equations and systems of linear equations. 

• Linear Equations
  Examines the relationship between functions and equations and solving systems of linear equations from the perspective of functions, both symbolically and graphically. The concept of equivalence is stressed.

• Systems of Linear Equations
  Explores problem-solving strategies, beyond the procedural, to solve systems of linear equations. The focus is on nonstandard approaches to solutions and on student perceptions and misperceptions of linear equations. 

Explores quadratic functions, equations, inequalities, and transformations through concrete situations and provides analysis of multiple ways to explore their solutions by viewing algebra as objects and processes. 

• Quadratic Functions
  Promotes the concept of quadratic functions as a central idea in working with quadratic equations to model real-world situations. This module stresses understanding student thinking about quadratic functions to inform practice and deepen their understanding.

• Quadratic Equations
  Connects physical, symbolic, tabular, and graphic representations of quadratic equations and functions and standard (textbook) and nonstandard solutions to these. This module stresses functions and relationships as the core of algebra.

• Transformations of Quadratic Functions
  Focuses on the transformations of graphic and symbolic representations of quadratic functions as a means for understanding the characteristics of and solving these functions.  In addition to transformations, this module emphasizes dilations, translations, and reflections of quadratic functions.

Focuses on Algebra II topics by presenting four areas of focus for professional learning: content knowledge, common student errors and misconceptions, pedagogical approaches and strategies, and actionable processes for gathering data and implementing effective lessons in the classroom.

• Operations on Numbers and Expressions
  Focuses on the critical foundational understanding of rational, real, and complex numbers, using both numeric and algebraic expressions, including expressions involving exponents and roots.

• Linear Equations and Inequalities for Algebra II
  Focuses on teaching equations and inequalities under the umbrella of functions, as well as making connections between symbolic, graphical, and other representations of equations and inequalities.

• Linear Systems for Algebra II
  Examines linear equations and inequalities in both two and three unknowns through the use of multiple representations and strategies for modeling and solving linear systems. This module identifies and unpacks common student errors that Algebra II students make in this content area.

• Quadratic Functions for Algebra II
  Focuses on the relationships among the graphical and symbolic forms of quadratic functions. This module highlights activities for manipulating the three symbolic forms of quadratic functions in order to inspect and predict the function’s shape, vertical orientation, and location.

• Transformations of Quadratic Functions for Algebra II
  Focuses on the transformations, dilations, translations, and reflections of quadratic functions. This module highlights common student errors and gaps in conceptual understanding as well as teaching strategies and intervention techniques to address student misconceptions.

• Exponential Functions
  Looks at functions as a way to understand the nature of exponential growth, the connections across different representations of exponential functions, and the connections between exponents and logarithms.

• Function Operations and Inverses
  Focuses on how and why we find inverses of functions. It goes beyond a procedural approach by helping teachers provide students with the rationale for why it is necessary to find inverses of functions. It emphasizes multiple representations and shows connections between compositions of functions, inverse functions, domain, range, and one-to-one comparisons.

• Higher Order Polynomials and Rational Functions
  Focuses on different characteristics of these functions, including characteristics of their algebraic representations (e.g., number of roots or zeros, odd or even, degree) and their graphical representations (e.g., vertical and horizontal asymptotes, holes and zeroes, behavior near various critical points).